Mathematics

Question

Verify the identity. cos quantity x plus pi divided by two = -sin x

2 Answer

  • [tex]\cos\left(x+\dfrac\pi2\right)=\cos x\cos\dfrac\pi2-\sin x\sin\dfrac\pi2=0\cos x-1\sin x=-\sin x[/tex]

    via the angle sum identity for cosine.
  • Answer with explanation:

    We are asked to prove the identity:

             [tex]\cos (x+\dfrac{\pi}{2})=-\sin x[/tex]

    We know that:

    [tex]\cos (A+B)=\cos A\cos B-\sin A\sin B[/tex]

    Here we have:

    [tex]A=x\ and\ B=\dfrac{\pi}{2}[/tex]

    Hence,

    [tex]\cos (x+\dfrac{\pi}{2})=\cos x\cos (\dfrac{\pi}{2})-\sin x\sin (\dfrac{\pi}{2})[/tex]

    We know that:

    [tex]\cos \dfrac{\pi}{2}=0\\\\and\\\\\sin \dfrac{\pi}{2}=1[/tex]

    Hence, we have:

    [tex]\cos (x+\dfrac{\pi}{2})=0-\sin x\\\\i.e.\\\\\cos (x+\dfrac{\pi}{2})=-\sin x[/tex]

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